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Edited by: Dipankar Chatterjee, Central Mechanical Engineering Research Institute (CSIR), India

Reviewed by: Wei Tang, National Institute of Standards and Technology (NIST), United States; Sandip Sarkar, Jadavpur University, India

This article was submitted to Thermal and Mass Transport, a section of the journal Frontiers in Mechanical Engineering

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Dynamic modes of fire propagation present a significant challenge for operational fire spread simulation. Current two-dimensional operational fire simulation platforms are not generally able to account for the complex interactions that drive such behaviors, and while fully coupled fire-atmosphere models are able to account for dynamic effects to an extent, their computational demands are prohibitive in an operational context. In this paper we consider techniques for extending two-dimensional fire spread simulators so that they are able to simulate certain dynamic fire behaviors. In particular, we consider modeling vorticity-driven lateral spread (VLS), which is characterized by rapid lateral fire propagation across steep, leeward slopes. Specifically, we consider modeling the influence of the fire on the local surface airflow via a “pyrogenic potential” model, which allows for vertical vorticity effects (in a near-field sense) using the Helmholtz decomposition. The ability of the resulting model to emulate fire propagation associated with VLS is demonstrated using a number of examples.

Fire spread simulators are an essential component in the assessment of wildfire risk. Given the requisite information on weather, topography and fuels, they provide fire management end-users with a way to map the likely evolution of an active wildfire across a landscape. Fire spread simulators can also be used to evaluate the effectiveness of different suppression options, as part of a technical assessment of individual fires, or they can be used to inform hazard reduction programs (e.g., prescribed burning or mechanical thinning) as part of broader strategic objectives. The effectiveness of a fire spread simulator, however, is critically dependent on: (i) the accuracy of the information that is used as its input; and (ii) the ability of the underpinning fire spread models and propagation algorithms to faithfully represent the main processes driving fire propagation. This second dependence becomes critical when a fire exhibits dynamic behaviors, which arise in response to multi-scale interactions between the fire and the local fire environment, namely the fuel, weather and topography.

In fact, the current suite of operational fire spread simulators (e.g., Phoenix Rapidfire, FARSITE) are poorly suited to modeling dynamic fire propagation. This is mainly due to their reliance on the assumption that a fire will spread at a quasi-steady rate uniquely determined by environmental conditions, and the assumption that different parts of a fire line propagate independently. This latter assumption, for example, is implicit in propagation algorithms such as those based on Huygens' Principle, which is often used in operational fire spread simulators (Finney,

Documented examples of dynamic fire propagation include that exhibited by junction fires (Viegas et al.,

Experimental fire in a wind tunnel showing a fire whirl (vortex) on the leeward slope of an idealized ridge. Note that the vortex is on the left flank of the fire and has components ω_{x}, ω_{z} < 0. The pyrogenic vorticity _{p} and its orientation are indicated in the figure. The figure has been adapted from Sharples et al. (

At present it is only possible to accurately model phenomena like VLS using three-dimensional coupled fire-atmosphere models. While such an approach is useful for providing insights into the physical processes that drive such behaviors, their computational cost makes them impractical for operational use. Sharples et al. (

In this paper we consider a recently developed approach to modeling fire spread (Hilton et al.,

We begin by giving a more detailed account of the VLS phenomenon in the next section, before outlining the model extension and its application in a number of specific examples.

McRae (

Subsequent investigation of the phenomenon by Simpson et al. (

Sharples et al. (^{−1}. In addition, VLS has been observed to occur almost exclusively in heavier fuels (e.g., forest fuels of the order of 15–20 t ha^{−1}). The conditions relating to topography and wind direction can be combined in a simple filter model that identifies parts of the landscape prone to VLS occurrence under a specified wind direction. The VLS filter takes the form of a binary variable, χ, which assumes a value of 1 in regions prone to VLS occurrence and 0 elsewhere. Mathematically, this can be expressed as follows:

Here

The parameters σ and δ, which define the VLS filter, represent threshold values for the topographic slope and θ, respectively. Only parts of the landscape with slopes greater than σ and with θ less than δ are prone to VLS. The values σ = 16° and δ = 40° were found to be appropriate for a digital elevation model of 90 m resolution, but may not be optimal for digital elevation models of different spatial resolution. While this is an important issue, which is currently the focus of ongoing research, it will not affect the results presented in the following sections.

While the VLS filter is useful for identifying slopes that are prone to VLS occurrence, laboratory experiments, wildfire observations and numerical simulations have revealed that the rapid lateral spread associated with VLS really only occurs in a relatively narrow portion of the leeward slope near the top of the hill (Quill and Sharples,

Unfortunately, the fact that VLS arises due to a strong coupling between the fire and the atmosphere, means that it is not possible to model VLS using existing two-dimensional fire spread simulators. These simulators, which are based on the notion of a quasi-steady rate of spread and the assumption that different points along a fire line can be treated essentially as independent source fires, are fundamentally unable to account for the dynamic interactions that drive VLS. While it is possible to model the VLS phenomenon using coupled fire-atmosphere models, their computational demand means that they are not feasible as operational tools. Hence, from the operational perspective, the possible effects of VLS on the overall propagation of a wildfire remain unresolved. Indeed, until computational resources evolve to the point that fully coupled fire-atmosphere simulations can be conducted in the order of minutes (rather than hours or days), there appears to be only two possible approaches to incorporating dynamic effects such as VLS in operational fire spread prediction:

Develop parameterizations of the dynamic behaviors, which then facilitate the use of specially tailored sub-models to emulate the observed behaviors; or

Develop reduced models that capture the main processes governing the dynamic behaviors, but which can be implemented in a highly computationally efficient manner.

Sharples et al. (

In the remainder of this manuscript we follow the second approach, and discuss a reduced model that accounts for pyroconvective coupling between the fire and the atmosphere in a very straightforward manner.

Hilton et al. (_{p} can be added to the ambient wind field, and this net wind field can be used to model the evolution of the fire. In the present work we use a level-set method to simulate the evolution of the fire perimeter, as implemented in the Spark fire simulation framework (Miller et al.,

To determine the pyrogenic flow _{p}, we invoke the Helmholtz Decomposition, which states that a twice continuously differentiable vector field with compact support can be expressed as the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field (Arfken and Weber,

Considering the flow _{p} induced by a fire, it is reasonable to assume that _{p} → 0 sufficiently far away from the fire. Hence if we make the assumption that _{p} is sufficiently smooth, we can then write:

for some scalar ψ and some vector

where ν = −∂_{z}_{z}, which represents the derivative of the plume updraft, and _{p} can be determined to account for the effects of the fire on the local atmosphere – we refer to these as

In particular, the model can be used to account for potential sources of vertical vorticity via the solenoidal term in (3), reducing the vector Poisson Equation (4) to:

and the resulting flow in the ground plane to:

The spread of a fire over a landscape can be modeled using a two-dimensional approach where the fire is represented as an interface between burnt and unburnt regions (Miller et al.,

where _{c}, plus a term depending on the wind field,

where _{a} + _{w}_{p}, where _{a} is an ambient wind vector, _{w} is the wind vector created by vorticity sources, given in Equation (6), and _{w} is an arbitrary constant governing the effect of the vortex-generated wind speed on the fire.

An example simulation using the pyrogenic vector potential coupled to a wildfire spread simulation is shown in _{c} = 0.5 and _{w} = 0.5 in Equation (8). These constants were chosen arbitrarily for illustration. The fire was started from a single start point of radius 4 m located 200 m in the horizontal and vertical directions away from a pyrogenic source term. This source term was a single point with ω_{z} = 5 at the indicated location. No ambient wind speed was used in the simulation with _{a} = 0. The resolution was set to 1 m and run for 200 s. The solid black lines show the position of the fire perimeter every 20 s time and the local wind vectors resulting from the pyrogenic model are shown as grayscale arrows. The effect of the vortex point source is to draw the fire perimeter in a circular path due to the resultant circulating flow around the source in the ground plane. The simulation took approximately 15 s to run on a NVidia GTX 1060 graphics processing unit.

Example application of the pyrogenic potential model with vortex source term to a dynamic wildfire simulation. The black lines show isochrones of a fire perimeter and the arrows show the resultant wind field from the vorticity source.

The key challenge in modeling VLS is to determine a way of translating the ambient horizontal vorticity that forms over a leeward slope due to flow separation, into vertical (pyrogenic) vorticity, ω_{z}. Specifically, we seek a closed-form solution for the components of the ambient horizontal vortex roll lofted by a buoyant fire plume.

The set-up under consideration is shown in

where

Schematic set-up of model.

We make the following assumptions for the flow dynamics on the lee slope:

(a) The flow can be approximated as steady state using the general steady-state inviscid vorticity equation as the outward spread of the fire is much slower than the wind flow. This is given by (Vallis,

where

(b) The dominant flow is the vertical lofting flow created by the fire plume; that is, _{z} ≫ _{x}, _{y}. This allows the _{x} and _{y} components to be neglected in Equation (11).

(c) The flow over the ridge results in a vortex that can be modeled as a prescribed source term. With no loss of generality, this can be aligned with the _{x} = 0. The source term is assumed to be a localized line source of the form:

where _{x} and Δ_{z} are the

(d) The assumption in the scalar model (Hilton et al., _{z}_{z}, is carried over so that _{z} = ν_{z} = 0 at

With these assumptions, Equation (11) reduces to:

and

Rewriting Equation (12) using assumptions (c) and (d) gives:

Equation (14) can now be solved for ω_{y} using Laplace transforms. The solution so obtained is:

where

Vorticity ω_{y} as a function of height

Rearranging Equation (13) gives:

and substitution of Equation (15) yields

This equation has an analytic solution of the form:

where

Equation (18) has the form of a ramp function starting at Δ_{z}. The solution supports a linear term _{z} source terms at _{y}_{z} ≠ 0. In the simplest possible case of _{z} → ∞ as _{z}_{z} and realistically Equation (18) only applies to regions below the free stream and above the flame source where the plume is accelerating.

Vorticity ω_{z} as a function of height

The pyrogenic model is applied at a nominal mid flame height _{0}. At _{0} + Δ_{z} and with

where:

This result can be generalized to the case of a line source given by a vector equation of the form _{xy}:

where _{z} within a localized region and _{z} = 0 outside the region ω_{z} will only be produced at the intersection of the line source _{xy} and the edges of the region. This will give rise to a source term where ∇_{z} · _{xy} > 0 and a sink where ∇_{z} · _{xy} < 0 resulting in two counter-rotating vortices, as illustrated in

Resultant vorticity in the

In this case the expression for the vertical vorticity can undergo a final simplification:

where

In this section we implement the two-dimensional model described in the previous section and evaluate its ability to capture the patterns of dynamic fire propagation associated with VLS. Moreover, we compare the performance of the two-dimensional model with output from a more sophisticated coupled fire-atmosphere model. To this end, we begin by giving a brief overview of the coupled modeling results.

Simpson et al. (^{−1} (at the surface) was allowed to flow over the hill, which had a windward slope angle of 20° and a leeward slope angles of 35°. A fire was initiated as a line ignition near the bottom of the leeward slope and allowed to spread. Full details of the simulations are provided by Simpson et al. (

The idealized simulations were conducted with the fire-atmosphere feedback turned off or turned on. When the fire-atmosphere feedback was turned off, the fire simply propagated back up the leeward slope toward the ridge line and spread laterally at a roughly uniform rate. An example of an uncoupled simulation can be seen in

Coupled fire-atmosphere model output of a fire burning on a leeward slope at times of 60, 90, and 120 min into the simulation. Panels

It is also important to note that the simulations conducted by Simpson et al. (

The pyrogenic potential model was implemented in the Spark framework, a software system for simulating wildfires (Hilton et al.,

For the purposes of the two-dimensional simulations the horizontal vorticity generated by the flow over the hill was assumed to be static and steady state. This is not a requirement of the model, but simplifies calculations as the assumption of a steady state vortex allows the backwards flow in the lee side of a hill to be imposed as a steady wind condition. The vertical vorticity is assumed to be the dominant component affecting the lateral spread of the fire in the ground plane and is dynamically calculated. The assumption reduced the vector Poisson equation (4) to the scalar Poisson Equation (5). The vertical vorticity in Equation (5) is calculated from Equation (23). The equation is numerically solved using a multigrid method (Hilton et al.,

A dynamic simulation under the idealized conditions given above is shown in ^{−1} on the windward slope. The re-circulation was prescribed by setting the wind speed to -1 ms^{−1} on the lee slope. The fire rate-of-spread, ^{−1} and a surface to volume ratio of 1159 ft^{−1}. The vorticity was prescribed as a line source with _{xy} = (0, 1) and

Dynamic calculation in Spark using idealized conditions of

The Dirac function was represented using a smoothed function:

where ϵ = 0.15 is a smoothing length scale, chosen to numerically smooth the Dirac function (Hilton et al.,

For the case with pyrogenic vorticity, left-hand side of

The pyrogenic potential model output compares favorably to the coupled fire-atmosphere model output. In particular, when the effects of pyrogenic vorticity are included, the two-dimensional model is able to produce patterns of fire propagation that are qualitatively similar to that produced by the coupled fire-atmosphere model (compare the left panel of

There are some notable differences between the two-dimensional model output and that of the fully coupled model. In particular, the lateral extent of the fire spread across the lower parts of the leeward slope, which are not prone to vorticity effects, is much less in the two-dimensional model output compared to that of the fully coupled model (even when the coupling is turned off). These differences are likely due to the influence of turbulence, which are not accounted for in the highly idealized two-dimensional pyrogenic potential model simulations.

Using the pyrogenic model imposed a modest computational overhead on the calculation. Using a NVidia GTX 1060 graphics processing unit the 2 h simulation took around 6 s to run with the pyrogenic vortex model and around 1 s without the model.

Dynamic modes of fire propagation arising from coupling between a fire and the atmosphere pose a significant challenge to two-dimensional fire spread simulators. Currently, such models are not able to accurately account for such behaviors. Here we have presented a new two-dimensional model based on a pyrogenic vector potential formulation that is able to reproduce a specific mode of fire-atmosphere interaction, namely, rapid lateral spread associated with VLS. The model accomplishes this by incorporating near-field effects driven by pyrogenic indrafts and local interaction of the fire with ambient horizontal vorticity. As such, the model can be seen as a “reduced physics” model, in which fire-atmosphere coupling has been greatly simplified. Despite these simplifications, however, the model is able to capture many of the key features observed in connection with VLS and other forms of dynamic fire spread (Hilton et al.,

The pyrogenic potential model has a significant computational advantage over the fully coupled fire-atmosphere models that have previously been required to accurately model VLS. The pyrogenic potential model took only about 10 s on a standard desktop computer to simulate 2 h of the spread associated with VLS, whereas the fully coupled model required around 8–10 h of to run on a current state-of-the-art high performance computing platform. This increase in computational efficiency could allow the model to be used in scenarios where computational speed is crucial, such as operational fire spread predictions.

Use of the pyrogenic model in operational prediction systems could provide fire managers with the ability to better appreciate the full range of fire behaviors that could be expected, especially under extreme conditions. For example, the VLS phenomenon has been associated with the generation of mass spotting events and the formation of deep flaming zones, which pose a serious threat to firefighter safety and can enhance the likelihood of pyrocumulonimbus development (McRae et al.,

Although the model presented here constitutes significant progress in our ability to efficiently model dynamic fire propagation, there are still further modeling scenarios that need to be considered, and a number of improvements that could be implemented. For example, the model has been shown to perform reasonably for only a single wind-terrain configuration. Other configurations such as those considered by Raposo et al. (

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation, to any qualified researcher.

The study was conceived and designed by both authors. JH conducted the numerical simulations and prepared the figures, with contributions from JS. The paper was written and reviewed by JS and JH.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors acknowledge the support of the Bushfire and Natural Hazards Cooperative Research Centre. Parts of the research presented was also undertaken with the assistance of resources and services from the National Computational Infrastructure (NCI), which is supported by the Australian Government.